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Theorem euen1b 8180
Description: Two ways to express "𝐴 has a unique element". (Contributed by Mario Carneiro, 9-Apr-2015.)
Assertion
Ref Expression
euen1b (𝐴 ≈ 1𝑜 ↔ ∃!𝑥 𝑥𝐴)
Distinct variable group:   𝑥,𝐴

Proof of Theorem euen1b
StepHypRef Expression
1 euen1 8179 . 2 (∃!𝑥 𝑥𝐴 ↔ {𝑥𝑥𝐴} ≈ 1𝑜)
2 abid2 2894 . . 3 {𝑥𝑥𝐴} = 𝐴
32breq1i 4793 . 2 ({𝑥𝑥𝐴} ≈ 1𝑜𝐴 ≈ 1𝑜)
41, 3bitr2i 265 1 (𝐴 ≈ 1𝑜 ↔ ∃!𝑥 𝑥𝐴)
Colors of variables: wff setvar class
Syntax hints:  wb 196  wcel 2145  ∃!weu 2618  {cab 2757   class class class wbr 4786  1𝑜c1o 7706  cen 8106
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-sep 4915  ax-nul 4923  ax-pr 5034  ax-un 7096
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 835  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-ral 3066  df-rex 3067  df-reu 3068  df-rab 3070  df-v 3353  df-sbc 3588  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4226  df-sn 4317  df-pr 4319  df-op 4323  df-uni 4575  df-br 4787  df-opab 4847  df-id 5157  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-res 5261  df-ima 5262  df-suc 5872  df-iota 5994  df-fun 6033  df-fn 6034  df-f 6035  df-f1 6036  df-fo 6037  df-f1o 6038  df-fv 6039  df-1o 7713  df-en 8110
This theorem is referenced by:  euhash1  13410  f1otrspeq  18074  hausflf2  22022  minveclem4a  23420
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